Question

Describe the transformations that bring g(x)−4x+1−−−−−√−3
𝑔
(
𝑥
)
−
4
𝑥
+
1
−
3
to h(x)=−234x+1−−−−−√−3
ℎ
(
𝑥
)
=
−
2
3
4
𝑥
+
1
−
3
.(1 point)
Responses

reflection over the line x=−3
𝑥
=
−
3
and a vertical compression by a factor of 32
3
2
reflection over the line x is equal to negative 3 and a vertical compression by a factor of 3 halves

reflection over the line y-axis and a vertical stretch by a factor of 32
3
2
reflection over the line y -axis and a vertical stretch by a factor of 3 halves

reflection over the line y=−3
𝑦
=
−
3
and a vertical compression by a factor of 23
2
3
reflection over the line y is equal to negative 3 and a vertical compression by a factor of 2 thirds

reflection over the line x=−3
𝑥
=
−
3
and a vertical compression by a factor of 23
2
3
reflection over the line x is equal to negative 3 and a vertical compression by a factor of 2 thirds

Answer 1

To describe the transformations that bring \( g(x) = \sqrt{-4x + 1} - 3 \) to \( h(x) = -\frac{2}{3} \sqrt{4x + 1} - 3 \), we need to analyze the components of both functions step by step.

1. **Initial function**:
\[
g(x) = \sqrt{-4x + 1} - 3
\]

2. **Identifying transformations**:
- The inside of the square root can be rewritten. The term \(-4x + 1\) can be rewritten as \(-1(4x - 1)\), showing a reflection over \(x\)-axis because of the negative coefficient of \(x\).
- The factor of \(-4\) can also be represented as a vertical stretch by a factor of 2 and then a reflection over the \(y\)-axis.

3. **Applying reflection and transformations**:
- Reflected over the line \(x = -3\) means if you further manipulate the form of the square root function.
- The \(-3\) outside the square root indicates a downward shift.

4. **Transformation reflecting into the new function**:
- The \( -\frac{2}{3} \) indicates a vertical compression (factor of \( \frac{2}{3} \)).
- The factor of \( 4x + 1 \) suggests a re-scaling combined with reflecting upon adjustments made.

Thus, the final transformation from \( g(x) \) to \( h(x) \) involves:

- **Reflection over the line y = -3**, leading to the negative factor outside the square.
- **A vertical compression by a factor of \( \frac{2}{3} \)**.

Therefore, the correct transformation description would be:
- **Reflection over the line \( y = -3 \) and a vertical compression by a factor of \( \frac{2}{3} \)**.